I've written on this before on r/AskLiteraryStudies so I'm doing a little copy/paste here with some edits. Hope it helps... Feel free to PM me if you want to continue the conversation.

It honestly depends on how you view the postmodern. It isn't as simple a thing as saying that there are signs and signals of a moving beyond it, because the theory of the post-modern sets up an unattainable goal which always-already encompasses and predicts everything that will come after. Depending on how much you want to delve into the theory side of things a successful argument for a post-postmodernism would have to address the paradox of the theory which sets an unattainable goal and explain how you can break free of those parameters. There are several attempts to do this in recent texts on the subject, but there is not an agreed upon departure from the postmodern in academia. If anything, the strongest movement to counter the postmodern is coming from the anti-postmodernists, who rather than try to deal with those issues would rather make claim that the postmodern never occurred. It is still somewhat of a fringe notion to have a post-postmodern, simply because there has not been an agreed upon consensus in the humanities or social science as to what that would be or what it looks like. Every attempt that I have researched could just as easily fit into the same modernist history and needs no addition of an extra post on the already troublesome postmodern. Jurgen Habermas, for one, sees the modernist project as being unfinished and thus the very concept of a postmodern as being ridiculous. His essay on this is called "Modernity - an Incomplete Project" and it makes a very convincing argument, while I don't totally agree with his attacks, he does make a very convincing argument.

If you are still interested in pursuing the concept here are some books on it that might help.

Alan Kirby has written a book called Digimodernism: How New Technologies Dismantle the Postmodern and Reconfigure Our Culture.

Raoul Eshelman's Performatism, or the End of the Postmodern offers a thesis on the topic. Semi-interesting, but I wasn't a huge fan.

Zygmunt Bauman's Liquid Modernity offers a great thesis on the changing aspects of the history of the modern.

N. Katherine Hales, How We Became Posthuman: Virtual Bodies in Cybernetics, Literature, and Informatics, offers in my view one of the best explanations for where the current pulse is at.

Nikolas Rose, The Politics of Life Itself, is a much denser argument and offers a very strong foundation for any such argument to be made.

Finally another book that I didn't care for but should be mentioned in this topic is by Gayatri Chakravorty Spivak, called An Aesthetic Education in the Era of Globalization. I like some the language she uses to discuss the topic, but I think that she has some very serious flaws in her argument...

I think we have to figure out what post-modernism really is first. I mean, there are basic tenants to postmodernism that you can read about on Wikipedia, and there's some art and literature and philosophy written about it...but to my eye it's all about the what and not the why. If we understood it we would give it a more descriptive name than "the movement that came after the other movement." So given that no one really knows what movement we're passing through now...I'm doubtful that we can give a name to the movement that will come later.

I've only begun my Psych courses but I'll throw in my two cents. Postmodernism seems broad and ill defined compared to the other schools of thought. Something needs to stand out to specifically title it as postmodernism. Perhaps in the future it will be properly labelled.